Blow-up results and localization of blow-up points in an \(N\)-dimensional smooth domain
Positive solutions of the heat equation are studied, on a bounded and smooth cylindrical domain \(\Omega\times ]0,T[\), subject to the conditions \(D_\eta u=f(u)\) on \(\partial\Omega \times ]0,T[\) and \(u(\cdot,0) =u_0\) on \(\Omega\). Here \(f\) is an increasing, positive, \(C^2\)-function such that \(\int^\infty (1/f) <\infty\), while \(u_0\) is a positive function in \(C^{2+\alpha} (\overline \Omega)\) such that \(D_\eta u_0 =f(u_0)\). An estimate for the blow-up time is established. Under the additional hypotheses that \(\Omega= B(0,1)\), \(f(0)=0\), and \(f\) is convex, it is also proved that the blow-up occurs at the boundary. For a general \(\Omega\), the same conclusion is shown to hold if the hypothesis \(\Delta u_0\geq c>0\) is also assumed. These results generalize earlier ones of several authors.
scientific article; zbMATH DE number 1042489